Question

The fish population in a local stream is decreasing at a rate of 3% per year. The original population was 48,000. Write an exponential decay function to find the population after 7 years. (Round to the nearest whole number).

a) P(t) = 48000(1 - 0.03)ᵗ; P(7) = 40652
b) P(t) = 48000(1 + 0.03)ᵗ; P(7) = 40652
c) P(t) = 48000(1 - 0.03)ᵗ; P(7) = 44769
d) P(t) = 48000(1 + 0.03)ᵗ; P(7) = 44769

Answer 1

Using the exponential decay function P(t) = 48000(1 - 0.03)t, we calculate that the fish population after 7 years is approximately 39182, when rounded to the nearest whole number.

Step-by-step explanation:

To find the population of fish after 7 years given an annual exponential decay rate of 3%, we use the exponential decay function P(t) = P0 × (1 - r)t, where P(t) is the population at time t, P0 is the original population, r is the decay rate, and t is time in years. In this scenario, P0 is 48,000, r is 0.03 (3%), and t is 7. Therefore, the correct function is P(t) = 48000(1 - 0.03)t.

Let's calculate the population after 7 years:

• P(7) = 48000 × (1 - 0.03)7
• P(7) = 48000 × 0.977
• P(7) = 48000 × 0.81629788
• P(7) = 39182 (rounded to the nearest whole number)

Hence, option (c) P(t) = 48000(1 - 0.03)t; P(7) = 44769 is closest to the calculated value but is not accurate. Therefore, a new option would be needed with the correct computation:

P(t) = 48000(1 - 0.03)t; P(7) = 39182 (rounded to the nearest whole number).